Consider the differential equation
xdydx−2y=0.
Find a second solution y=Cxn
that might not be a general solution and which may have a different value of n
than your first solution.

Given the differential equation:

\[
x \frac{dy}{dx} - 2y = 0
\]

We need to find a second solution of the form \( y = Cx^n \), where \( C \) is a constant and \( n \) is to be determined.

### Step 1: Substitute \( y = Cx^n \) into the differential equation

First, we calculate \( \frac{dy}{dx} \) for \( y = Cx^n \):

\[
\frac{dy}{dx} = Cn x^{n-1}
\]

Substituting \( y = Cx^n \) and \( \frac{dy}{dx} = Cn x^{n-1} \) into the differential equation:

\[
x(Cn x^{n-1}) - 2(Cx^n) = 0
\]

### Step 2: Simplify the equation

Distribute and simplify:

\[
Cn x^n - 2Cx^n = 0
\]

Factor out \( Cx^n \) from the equation:

\[
Cx^n (n - 2) = 0
\]

Since \( Cx^n \) cannot be zero for all \( x \) (except in the trivial case where \( C = 0 \), which we disregard since we want a non-trivial solution), we have:

\[
n - 2 = 0
\]

Thus,

\[
n = 2
\]

### Conclusion

The second solution is:

\[
y = Cx^2
\]

This solution has \( n = 2 \), which is different from \( n = 1 \) that could have been obtained if we directly solved the original differential equation (by separation of variables or another method), which typically leads to a general solution in the form \( y = C_1 x^2 \).

### Do you have any further questions or would you like more details?

---

Here are five related questions to consider:

1. How would you solve the differential equation using separation of variables?
2. Can you find a general solution for the given differential equation?
3. What is the significance of finding a particular solution in the form \( y = Cx^n \)?
4. How does the solution change if we change the coefficient from \( -2y \) to another constant?
5. What happens to the solution if \( y \) is a more complex function, such as a polynomial of higher degree?

**Tip:** When dealing with differential equations, verifying solutions by substituting them back into the original equation is a good practice to ensure correctness.

Explore how to find a second solution to the differential equation x(dy/dx) - 2y = 0 using the method of undetermined coefficients. Learn about the general form of solutions in differential equations and the significance of different values of n in the solution. Suitable for undergraduate students studying differential equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation